I'm interested in the finite dimensional homogeneous spaces of $Diff(S^1)$. There are several papers concerned with infinite dimensional homogeneous spaces such as $Diff(S^1)/S^1$, but I can't find anything of the finite dimensional spaces, except the paper by Cartan from 1905, as hinted by Robert Bryant in the above link... From R.B's reply in the link:

"You could try É. Cartan's papers on infinite pseudogroups (mostly appearing 1904-05). In particular, see Paragraph 57 of Sur la structure des groupes infinis de transformation (suite). There, for example, he proves that the (pseudo-)group of diffeomorphisms of the line has three distinct (i.e., nonequivalent) 2-dimensional homogeneous spaces and seven distinct 3-dimensional homogeneous spaces, etc."

Basically I'm interested in the two dimensional spaces... it's just that I don't know French, and the exposition in Cartan's paper seems quite brief anyway. Surely there must be more modern works about the subject?

Also, it's not very clear to me how to construct these homogeneous spaces as cosets...

EDIT: Here's an example from Cartan's 1905 paper (I still don't know French but I was able to decipher that much):

Suppose we have vector fields as a Lie algebra of $Diff(\mathbb R)$ (Cartan is doing it on the line, but the circle is similar),

$l_n = x^{n+1} \partial_x$.

Cartan obtained corresponding vector fields on three dimensional homogeneous spaces of $Diff(\mathbb R)$. On one of the seven homogeneous spaces they are

EDIT: Fixed a misunderstanding regarding Cartan's notation in the formula (see RB's answer below). I expanded the vector fields in a basis by Taylor expanding the function $f(x) = \sum c_n x^{n+1}$. That way it's easy to see that they both satisfy the Witt algebra, $[l_n, l_m] = -(n-m)l_{n+m}$...

1 Answer
1

I'm afraid that you are mistranslating Cartan's notation. The first 3-dimensional example that Cartan gives is this: Every diffeomorphism of the line, written in the form $X = f(x)$, can be 'lifted' to a diffeomorphism of 3-space:
$$
X = f(x),\qquad Y = \frac{y}{f'(x)},\qquad Z = \frac{z}{f'(x)} - \frac{f''(x)}{f'(x)^2}\ .
$$
These diffeomorphisms act transitively on the complement of the plane $P$ defined by $y=0$. This defines a homomorphism from $\textrm{Diff}(\mathbb{R}^1)$ to $\textrm{Diff}(\mathbb{R}^3\setminus P)$ that is Cartan's first example of a 3-dimensional homogenous space of $\textrm{Diff}(\mathbb{R}^1)$. The corresponding homomorphic lifting of vector fields is given by
$$
\Phi\left(h(x)\frac{\partial\ }{\partial x}\right)
= h(x)\frac{\partial\ }{\partial x} - h'(x)y \frac{\partial\ }{\partial y}
- \bigl(h'(x)z + h''(x)\bigr)\frac{\partial\ }{\partial z}
$$
(and not the formula that you gave above).

This particular example is actually a fibered product of two $2$-dimensional homogeneous spaces of $\textrm{Diff}(\mathbb{R}^1)$, namely, the actions
$$
X = f(x),\qquad Y = \frac{y}{f'(x)}\ .
$$
(you should think of this $\mathbb{R}^2$ as the bundle of $1$-forms on the line) and
$$
X = f(x),\qquad Z = \frac{z}{f'(x)} - \frac{f''(x)}{f'(x)^2}
$$
(you should think of this $\mathbb{R}^2$ as the bundle of $0$-jets of affine connections on the line), fibered over their common $1$-dimensional homogeneous space $\mathbb{R}^1$, with the original action $X = f(x)$.

Three of the seven $3$-dimensional homogeneous spaces of $\textrm{Diff}(\mathbb{R}^1)$ listed by Cartan are fibered products of a pair drawn from the three $2$-dimensional homogeneous spaces of $\textrm{Diff}(\mathbb{R}^1)$ in this way, and three of them are the spaces of $1$-jets of sections of the $2$-dimensional homogeneous spaces, when they are regarded as bundles of rank $1$ over $\mathbb{R}^1$ (the $1$-dimensional homogeneous space). That leaves only one $3$-dimensional homogeneous space 'unexplained' in some geometric way. (In Cartan's list, this is the sixth example. It is some kind of natural bundle over the bundle of $0$-jets of affine connections on the line, but I don't yet see a simple way to describe it.)

Cartan's exposition is brief, as you say. This is typical of Cartan; after all, he is only working out an example of the general theory. I'm not aware of any place in the modern literature that describes this particular classification, but, on the other hand, once you know how to read Cartan, his argument fairly straightforward to follow.

If there is interest, I can sketch out a paraphrase of Cartan's argument in currently fashionable language.